Um estudo de curvas e superfícies sobre variedades diferenciáveis e o teorema de Hartman-Grobman
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Data
2023-07-04
Autores
Gutierrez Ccari, William
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Universidade Estadual Paulista (Unesp)
Resumo
Neste trabalho nosso primeiro objetivo foi estudar a teoria local das curvas (ou seja, propriedades que dependem apenas do comportamento da curva nas proximidades de um ponto) e superfícies regulares, ambas no espaço euclidiano três dimensional. O estudo de curvas e superfícies foi para entender as noções de variedades diferenciáveis, que de certa forma são uma generalização de superfícies em todas as suas dimensões. Como nosso segundo objetivo foi estudar a teoria qualitativa das equações diferenciais ordinárias para entender, demonstrar e dar alguns exemplos do Teorema de Hartman- Grobman. O Teorema de Hartman-Grobman garante que, se temos uma equação diferencial ordinária não linear e ela tem um ponto crítico hiperbólico então, podemos entender o comportamento qualitativo de suas soluções em uma vizinhança do ponto crítico hiperbólico, e isso observando o comportamento das soluções de sua equação diferencial ordinária linear associada.
In this work, our first objective was to study the local theory of curves (that is, properties that depend only on the behavior of the curve in the vicinity of a point) and regular surfaces, both in three-dimensional Euclidean space. The study of curves and surfaces was to understand the notions of differentiable manifolds, which in a way are a generalization of surfaces in all their dimensions. As our second objective was to study the qualitative theory of ordinary differential equations to understand, demonstrate and give some examples of the Hartman-Grobman Theorem. The Hartman-Grobman Theorem guarantees that if we have a nonlinear ordinary differential equation and it has a hyperbolic critical point then we can understand the qualitative behavior of its solutions in a neighborhood of the hyperbolic critical point, and this by observing the behavior of the solutions of its associated linear ordinary differential equation.
In this work, our first objective was to study the local theory of curves (that is, properties that depend only on the behavior of the curve in the vicinity of a point) and regular surfaces, both in three-dimensional Euclidean space. The study of curves and surfaces was to understand the notions of differentiable manifolds, which in a way are a generalization of surfaces in all their dimensions. As our second objective was to study the qualitative theory of ordinary differential equations to understand, demonstrate and give some examples of the Hartman-Grobman Theorem. The Hartman-Grobman Theorem guarantees that if we have a nonlinear ordinary differential equation and it has a hyperbolic critical point then we can understand the qualitative behavior of its solutions in a neighborhood of the hyperbolic critical point, and this by observing the behavior of the solutions of its associated linear ordinary differential equation.
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Palavras-chave
Curvas, Superfícies, Variedades diferenciáveis, Equações diferenciais ordinárias, Teorema de Hartman-Grobman, Curves, Surfaces, Differentials manifolds, Ordinary differential equations, Hartman-Grobman theorem